Ground states for pseudo-relativistic Hartree equations of critical type

TL;DR

This paper investigates the existence of ground state solutions for a class of non-linear pseudo-relativistic Schrödinger equations with critical two-body interactions, employing variational methods and boundary condition transformations.

Contribution

It introduces a novel approach to analyze pseudo-relativistic equations with critical interactions using variational techniques and boundary condition transformations.

Findings

Existence of ground state solutions established

Application of variational methods to nonlocal operators

Transformation to elliptic equations with nonlinear boundary conditions

Abstract

We study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schr\"odinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely related to the square-root of the Laplacian. We investigate such a problem using variational methods after transforming the problem to an elliptic equation with a nonlinear Neumann boundary conditions.